Thin-walled Cross-section Research Articles

Dynamic Collapse of Thin-Walled Composite Members -Characteristic Evaluation by Axial Compression and Bending Tests-

Many thin wall closed section members made of steel are used for the traffic machine such as an automobile. In these members, the local buckling under compressive stress is mentioned as a main problem. As a method for improving the performance, the technique which fills the inside of closed section with the low-density foaming material is used. In this study, axial compression and 3-point bending tests were carried out on the composite material which filled 2 kinds of epoxy resin foaming material in which the foaming rate (density) is different for hat type thin-walled cross section member. The difference of composite member and thin-walled member was investigated on elasticity and plastic collapse characteristics at the various deformation velocities. The adhesive strength between thin-walled member and foaming material, and filling effect of the foaming material in the energy absorption characteristics was clarified.

Static analysis of tapered FRP transmission poles using finite element method

This paper deals with linear static analysis of tapered FRP (fibre-reinforced polymers) transmission poles with circular thin-walled cross-section using a second order shell element and first order shear deformation theory (FSDT). The cross-section is treated as a generally orthotropic laminate. Typical poles are analyzed using a provided computer code by MATHEMATICA program. The results obtained by presented numerical modeling are verified through comparing to analytical results or those obtained from ANSYS 11.0 commercial finite element program. The effects of various parameters such as fibre orientation and type, fibre volume fraction, number of layers and geometry on the behavior of tapered FRP transmission poles are investigated.

A constrained finite strip method for open and closed cross-section members

In this paper, the constrained finite strip method (cFSM) is improved by simplifying the derivation of the constraint matrix which defines the combined buckling mode space formed by global and distortional modes. In the context of the FSM, only two assumptions are made to define this particular and complex subspace: (i) cylindrical plate bending (if any) and (ii) negligible in-plane transverse strains and shear strains. The constraint matrix is derived by means of procedures commonly used in numerical methods, such as FEM or FSM. This led to a simple and systematic formulation which allowed the cFSM to compute pure buckling modes for members with open and closed thin-walled cross-sections.

Vibration Analysis of Composite Beams With End Effects via the Formal Asymptotic Method

Vibration analysis of composite beams is carried out by using a finite element-based formal asymptotic expansion method. The formulation begins with three-dimensional (3D) equilibrium equations in which cross-sectional coordinates are scaled by the characteristic length of the beam. Microscopic two-dimensional and macroscopic one-dimensional (1D) equations obtained via the asymptotic expansion method are discretized by applying a conventional finite element method. Boundary conditions associated with macroscopic 1D equations are considered to investigate the end effect. It is then described how one could form and solve the eigenvalue problems derived from the asymptotic method beyond the classical approximation. The results obtained are compared with those of 3D finite element method and those available in the literature for composite beams with solid cross section and thin-walled cross section.

Effect of Bracing on Linear Free Vibration Characteristics of Thin-Walled Beams with Open Cross Section

The paper presents an analysis of the coupled vibration of beams with arbitrary thin-walled open cross section, braced with identical transversal header beams uniformly distributed along their length. The explicit form of analytic solution is derived by directly solving the governing differential equations of motion. The development is based on Vlasov theory which includes the effect of flexural-torsion coupling, the constrained torsion warping, and rotary inertia. The governing differential equations for coupled bending-torsional vibrations are performed using the principle of virtual displacements. In the case of simply supported beam, exact explicit expressions are derived to predict the natural frequencies and the corresponding mode shapes. The frequency equation, given in determinantal form, is expanded in an explicit analytical form, and then solved using the symbolic computing package Mathcad. The expressions are concise and very simple and as such convenient to be used by a practicing engineer who does not need to go into detail of thin-walled beam theory. Also, the use of explicit expressions gives significant savings in computing time compared with the alternative numerical methods [finite-element method (FEM), finite strip method, differential transform method, etc.]. To demonstrate the validity of this method the natural frequencies of braced thin-walled beams, having coupled deformation modes, are evaluated and compared with FEM.

Actuation of Slender Thin-wall Anisotropic Open Cross-section Beams Based on Asymptotically Correct Vlasov Theory

An asymptotically correct analysis of passive anisotropic thin-walled open cross-section beam-like structures using the variational asymptotic method (VAM) is extended to include embedded macro fiber composites. Application of the VAM to beam-like structures splits the problem into non-linear 1D theory along the selected beam reference line and linear 2D generalized 5 × 5 Vlasov theory augmented by a 5 × 1 actuation vector over the cross-section. The linear 2D cross-sectional theory is validated against the University of Michigan/variational beam sectional analysis 2D finite element software. The validation examples selected were based on practical cross-sectional geometry and material anisotropy under DC actuation voltage. Actuation-induced deformations predicted at the beam reference line are obtained using an intrinsic geometrically exact beam theory for open cross-sections. The predicted generalized deformations are compared with those obtained using the 3D finite element analysis software ANSYS Multiphysics, which further validates the extended theory. The analytical theory is shown to be straightforward to implement and efficient, yet sufficiently reliable to perform interdisciplinary studies and optimization of various engineering applications of such structures.

Transient dynamic analysis of tapered FRP composite transmission poles using finite element method

Power transmission poles are subjected to dynamic cantilever bending due to wind gusts and cable unilateral failure, or may also be subjected to vehicle impacts. In this paper, transient dynamic analysis of tapered fiber-reinforced polymer (FRP) composite transmission poles with circular thin-walled cross-section subjected to dynamic cable tension and vehicle impacts is investigated by combination of tapered beam finite element and precise time integration method. It is assumed that the material behavior is linearly elastic and the laminate of the cross-section of the wall is symmetric or antisymmetric angle-ply. The effect of fiber type and orientation, the pole geometry, and the concentrated mass at the pole tip are evaluated by performing the dynamic analysis of FRP poles under step, triangular and sine pulses. There is a good agreement between the results of the present method and those obtained from the poles modeled by ANSYS commercial finite element software and existed literatures. Also, there is a significant shorter run-time in the present method. It is concluded that beyond 10 layers for the laminate with constant thickness of the wall, the pole tip deflection does not change.

Local Web Buckling of Composite (FRP) Beams

Local buckling analysis of thin-walled open or closed section fiber-reinforced plastic beams is presented. In the analysis, the web is modeled as a long orthotropic plate with rotationally restrained edges. Explicit expressions were developed for the buckling analyses of rectangular (long) plates in a companion paper. These results are applied to develop explicit expressions for the calculation of the web buckling of beams with thin-walled cross sections. At last, the applicability of the method is demonstrated by numerical examples and the results are verified by finite element calculations.

Strength Evaluation of Cold-Formed Steel Columns Using The Results of Finite Strip And Finite Element Linear Stability Analysis

Most cold-formed steel columns display open and rather thin-walled cross-sections which mean that their structural behaviour is strongly affected by local and global buckling. Th e local mode, that occurs for shorter profi les, is characterized by (i) the local plate mode (LPM) characterized by the simultaneous flexural buckling of the web and fl anges and (ii) by the distortional mode (DM) characterized by the displacements of flange-stiff ener edges (that remain plane). The global mode occurring for long profi les is characterized by (i) the fl exural mode (FM) characterized by the translation of the whole section in the direction of the major principal axis and (ii) by the fl exural-torsional mode (FTM) characterized by the simultaneous translation and rotation of the whole section. Th e possibility of using the results of linear stability analysis in the national codes of thin-walled cold-formed steel structural elements (for instance, European and Brazilian Codes) arises, i.e. local and global buckling instability modes and corresponding bifurcation stresses determining the ultimate strength of members. Two powerful numerical methods are chosen to perform a linear stability analysis of a cold-formed steel structural member: (i) the Finite Strip Method, (i1) the Semi-Analytical Finite Strip Method (trigonometric functions are used in the approximation of displacement) used for simply supported boundary conditions, (i2) the Spline Finite Strip Method (‘spline’ functions are used in the approximation of displacement) used other boundary conditions and (ii) the Finite Element Method. The linear local and global stability results of for Z, C and rack cold-formed columns are used to obtain ultimate strength through the procedures adopted in the Eurocode 3, Part 1.3 and in the Brazilian Code (NBR 14.762/2001). The obtained numerical estimates by specifi cations are compared with experimental results available in literature.

Vibration Control of Composite Spacecraft Booms Subjected to Solar Heating

In this paper, piezoelectric feedback control of vibration and instability of spacecraft booms modeled as circular thin-walled cross-section beams and subjected to solar radiant heating is investigated. Having in view that composite material systems are likely to play a great role in the design of these devices, the beam constituent materials encompass non-classical effects such as anisotropy and transverse shear. In addition, in order to induce beneficial elastic couplings, a special ply-angle distribution achieved via the usual helically wounding fiber-reinforced technology, the so called filament winding, is implemented. The dynamic governing equations including the temperature effects and the related boundary conditions are obtained via the application of Hamilton's principle. Toward the end of controlling the oscillations and prevent the occurrence of the thermal dynamic instability, a feedback control capability based on the use of the piezoelectric induced strain actuation is implemented. The performance of its implementation considered in conjunction with that of the structural tailoring are highlighted and pertinent conclusions are derived.

Thermoelastic lateral-torsional buckling of fixed slender beams under linear temperature gradient

The thermal expansions and rotations that result from a linear in-plane temperature gradient field are fully restrained at the ends of a fixed beam. These restrained expansions and rotations will produce internal bending and compressive actions in the beam, and these actions increase with an increase of the temperature differential and average temperature of the linear temperature gradient field. When these actions reach critical values, the fixed beam may bifurcate from its primary equilibrium state to a buckled equilibrium configuration. This paper presents a systematic treatment of classical buckling analysis for thermoelastic lateral-torsional buckling and for in-plane thermoelastic flexural buckling of a fixed beam of doubly symmetric open thin-walled cross-section that is subjected to a linear temperature gradient field over its cross-section. It is shown that the effective centroid and shear centre, rather than the geometric centroid and shear centre, should be used in formulating the thermoelastic prebuckling and buckling analysis and that the effects of temperature on the buckling resistance need to be considered. The thermoelastic lateral-torsional buckling of a fixed beam under a linear temperature gradient field is more complicated than its mechanical counterpart for uniform bending or for uniform compression, and iterative methods are needed to obtain accurate solutions.

An asymptotic analysis of composite beams with kinematically corrected end effects

A finite element-based beam analysis for anisotropic beams with arbitrary-shaped cross-sections is developed with the aid of a formal asymptotic expansion method. From the equilibrium equations of the linear three-dimensional (3D) elasticity, a set of the microscopic 2D and macroscopic 1D equations are systematically derived by introducing the virtual work concept. Displacements at each order are split into two parts, such as fundamental and warping solutions. First we seek the warping solutions via the microscopic 2D cross-sectional analyses that will be smeared into the macroscopic 1D beam equations. The variations of fundamental solutions enable us to formulate the macroscopic 1D beam problems. By introducing the orthogonality of asymptotic displacements to six beam fundamental solutions, the end effects of a clamped boundary are kinematically corrected without applying the sophisticated decay analysis method. The boundary conditions obtained herein are applied to composite beams with solid and thin-walled cross-sections in order to demonstrate the efficiency and accuracy of the formal asymptotic method-based beam analysis (FAMBA) presented in this paper. The numerical results are compared to those reported in literature as well as 3D FEM solutions.

Dynamic behavior of braced thin-walled beams

The present paper considers the dynamic behavior of beams with open thin-walled cross-sections along their length with bracings (connecting beams and truss). The effect of the constrained torsion warping, rotary inertia, and flexural-torsion coupling due to nonsymmetric cross-sections is included. In the case of simply supported beams, closed-form solutions for determining coupled natural frequencies and corresponding mode shapes are newly derived. The frequency equation, given in a determinantal form, is expanded in an explicit analytical form and then solved using the Mathcad 2001 Professional symbolic computing package. Some illustrative examples on the application of the present theory are given for coupled bending-torsion vibrations of braced thin-walled beams. As compared with FEM, numerical results demonstrate the accuracy and effectiveness of the proposed method

New Forming Processes for Sheet Metal with Large Plastic Deformation

Due to the high effort involved, bifurcated constructions in mass market products made from sheet metal remained largely unused. Extruded profiles with cross-sections containing bifurcations show the possibility to increase the stiffness and allow modern lightweight design using load optimized structures as well as in box strap, sandwich and stringer constructions or different profiles. The two new forming processes linear flow splitting and linear bend splitting developed at the PtU enable the production of bifurcated profiles in integral style made of sheet metal without joining, lamination of material or heating of the semi-finished product. These forming processes use obtuse angled splitting rolls and supporting rolls to transform the sheet metal at ambient temperature. Whereas the linear flow splitting process increase the surface of the band edge and forms the band into two flanges. At linear bend splitting a bended sheet metal as semi finished product is used. Thereby bifurcations at nearly any place of a sheet metal can be produced. Both processes induce high hydrostatic compressive stresses in the local forming zone during the process which leads to an increased formability of the material and thereby to the realization of large strains. Parts produced are characterized by increased stiffness, high surface hardness and low surface roughness. Experimental investigations have shown an increasing of the band edge surface at maximum splitting depth up to 1800%. By a following forming process new multi-chambered structures and integral stringer construction can be realized with thin walled cross-sections from steel of higher strength.

An analytical method for calculating torsional constants for arbitrary complicated thin-walled cross-sections

In this paper, an analytical method is proposed for calculating torsional constants for complicated thin-walled cross-sections with arbitrary closed or open rib stiffeners. This method uses the free torsional theory and the principle of virtual work to build governing equilibrium equations involving unknown shear flows and twisting rate. After changing the form of the equations and combining these two unknowns into one, torsional function, which is a function of shear flow, shear modulus, and twisting rate, is included in the governing equations as only one of the unknowns. All the torsional functions can be easily obtained from these homogeneous linear equations, and torsional constants can be easily obtained from the torsional functions. The advantage of this method is that we can easily and directly obtain torsional constants from the torsional functions, rather than the more sophisticated shear flow and twisting rate calculations. Finally, a complicated thin-walled cross-section is given as a valid numerical example to verify the analytical method, which is much more accurate and simpler than the traditional finite element method.

A full modal decomposition of thin-walled, single-branched open cross-section members via the constrained finite strip method

This paper derives a new method for fully decomposing the elastic stability solution, of a thin-walled single-branched open cross-section member, into mechanically consistent buckling classes associated with global, local, distortional, and shear and transverse extension buckling modes. The method requires a set of formal mechanical definitions for each of the buckling classes. For global and distortional buckling the definitions employed successfully by generalized beam theory are utilized herein, while for local and other (shear and transverse extension) buckling, new definitions are provided. The mechanical definitions for a given buckling class represent a series of constraint conditions on the general deformations that the thin-walled cross-section may undergo. These constraint conditions are derived as explicit constraint matrices within the context of the finite strip method, and provide the desired decomposition of the buckling deformations of the member. The decomposition is full in the sense that the union of the deformation spaces of the decomposed buckling classes is the same as the general deformation space in the original finite strip method. The resulting method is termed the constrained finite strip method (cFSM). The two primary applications for cFSM are modal decomposition and modal identification. Modal decomposition reduces the general finite strip solution to a desired set of buckling classes and performs a useful model reduction that allows the results to focus on a particular buckling class, e.g., distortional buckling. Modal identification provides a means to quantify the extent to which a given buckling class is contributing to a general buckling deformation. Application of cFSM, including graphical representation of the buckling classes, and the advantages of modal decomposition and modal identification, are provided in a series of numerical examples.

Local buckling of wide-flange thin-walled anisotropic composite beams

The present contribution deals with the onset of local buckling of compressively loaded thin-walled beams with open I, C, Z, T and L-cross-sections made of laminated composite materials. The method employs a discrete plate analysis approach in the course of which each structural subelement of interest—which presently is the flange—of the thin-walled cross-section is considered as a separate composite plate with elastic rotational restraints at those edges where an adjacent substructural element is located. While in many investigations the lamination schemes of webs and flanges are considered to be purely orthotropic, in the present paper the laminate layups are allowed to be of an arbitrary non-orthotropic nature, which also allows for the analysis of laminates with inherent bending–torsion coupling. The analysis of the buckling loads of the flanges of thin-walled composite beams is performed using the Ritz-method for which some especially adjusted displacement shape functions are employed. For the case of pure orthotropy, a novel closed-form solution is described. The accuracy of the employed approaches is established by comparison with accompanying finite element simulations of thin-walled composite beams. It is revealed that the presented methodology is highly efficient in terms of computational effort and yet performs with satisfying accuracy, which makes it very attractive for actual practical applications whenever the local stability behaviour of wide-flange thin-walled composite beams is to be considered.

5422 予備設計段階における薄板構造の断面形状の最適化(G12-1 設計と最適化,G12 設計工学・システム)

In this paper, we present a numerical method to optimize the cross section shape of thin-walled structures, which is often needed at the preliminary stage in the structural designs of automotives and airplanes and so on. Both open and closed cross section shapes are optimized. X and y coordinate values of all the nodes defining the shape are set as design variables. Optimized or identified shape is determined by combining some response values of sectional properties such as area, moments of inertia, torsional constant and shear center. The problem is solved using the sequential quadratic programming method. The validity of this method is verified for obtaining the optimal shape of thin-walled cross section through examples.

Doubly coupled vibration band gaps in periodic thin-walled open cross-section beams

This paper studies flexural-torsional doubly coupled vibration through periodic thin-walled open cross-section beams composed of two kinds of materials. Based on the doubly coupled vibration equation, plane wave expansion method for the thin-walled beams is provided. If the filling fraction keeps constant, the lattice is one of the factors that affect the normalized gap width. If the lattice and filling fraction keeps constant, the Young modulus contrast plays a fundamental role for the band gap width, but not density contrast. Finally, the frequency response of a finite periodic binary beam is simulated by finite element method, which provides an attenuation of about 40dB in the frequency range of the band gaps. The findings will be significant in the application of phononic crystals.

Nonlinear analysis of members curved in space with warping and Wagner effects

The centroidal axis of a member that is curved in space is generally a space curve. The curvature of the space curve is not necessarily in the direction of either of the principal axes of the cross-section, but can be resolved into components in the directions of both of these principal axes. Hence, a member curved in space is primarily subjected to combined compressive, biaxial bending and torsional actions under vertical (or gravity) loading. In addition, warping actions in particular may occur in curved members with an open thin-walled cross-section, and as the deformations increase, significant interactions of the compressive, biaxial bending and torsional actions occur and profoundly nonlinear deformations are developed in the nonlinear range of structural response. This makes the nonlinear behaviour of a member curved in space very complicated, making it difficult to obtain a consistent differential equation of equilibrium for the nonlinear analysis of members curved in space. In addition, because torsion is one of the primary actions in these members, when the torsional deformations become large, the Wagner effects including both Wagner moment and the conjugate Wagner strain terms are increasingly significant and need to be included in the nonlinear analysis. This paper takes advantage of the merits of so-called “geometrically exact beam theory” and the weak form formulation of the differential equations of equilibrium in beam theory, and it develops consistent differential equations of equilibrium for the nonlinear elastic analysis of members curved in space with warping and Wagner effects. The application of the nonlinear differential equations of equilibrium to various problems is illustrated.